TPT November 2010

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114 November 2010 ›
Article
Application of the stereology
reconstruction methods in assessment
of the spatial grain structure of
metals and alloys
V V Perchanik, Ye Ya Lezinskaya, D Yu Klyuev (National Metallurgical Academy of Ukraine)
N A Koryaka (ITA Representative in CIS, Ukraine)
Distribution of grain sizes in the volume of a metal product is an
important characteristic of dispersivity and homogeneity, and hence
of properties and endurance of the product and the entire structure
during its operation.
Depending on the application of the product and the conditions
of its using, the requirement to a grain size and variation in grain
size is a criterion of stability and reliability of this product. For
example, requirements to nuclear power plant (NPP) fuel element
cladding tubes, tubes for bellows, capillary tubes used as heaters in
incandescent lamps, etc.
Due to opacity of metals, a specially treated flat cut (a polished
section) offers an initial information about the structure, which allows
determination of the averaged grain size in the examined plane using
reliable existing standard methods (GОST 5639, ASTM E112, etc).
A large number of methods of structure reconstruction by its flat image
have been developed since the 1930s, because the flat cut is just an
indirect reflection of the spatial metal structure which is responsible for
all physical and mechanical properties of metal products.
Spherical shape of structural components was the basis of all
developed methods of reconstruction.
Description of the known methods of reconstruction of the
stereological objects by their mapping on a plane has been
thoroughly made in a paper [1] which shows that all methods of the
structure stereology reconstruction by its mapping can be reduced to
solution of an integral equation which characterises the probabilistic
relation between mapping parameters and the actual size of circular
or spherical elements of the statistical population.
Our secondary analysis of a number of methods based on distribution
of the chord lengths in a random section has shown that they were
erroneous because of an incorrectly determined measure of the
geometric probability elements.
The formulas of Spector, Bocstiegel, Lord and Willis correspond
exactly to each other after a number of their transformations, and
the calculations made in accordance with these formulas confirm
experimentally the authors’ mistake because reconstruction
results in the structure refining which is physically inexplicable.
Nevertheless, the formula of Bocstiegel was used widely in the
programs of quantitative evaluation of the metal structure with the
use of Epiquant and Quantimet microscopes.
More correct methods are those based on the distribution of
random sections of diameters of spherical objects (eg Scheil’s
method developed in 1931 and improved later by Schwartz and
Saltykov [2]). The experimental statistics of distribution of ‘diameters’
of the flat cut circles (maximum sizes of each grain) is the base of
this method.
This method is quite correct from the point of view of choice of
the geometric probability measure but it does not provide a strict
mathematical procedure of reconstruction and has a number of
restrictions like the following:
• predetermination of the discrete testing intervals in the general
volume of the statistical population;
• method for derivation of source data and a compulsory account
for all elements of distribution of objects in the flat cut;
• poorly representing statistics and an intricate calculation method
with successive substitutions and accumulation of errors need
a radical improvement which does not allow this method to be
widely used in industry.
Attempts made by Schwartz and Saltykov to ‘improve’ the Scheil’s
method have not practically changed this method essence and were
unsuccessful. The use of Saltykov’s function of inverse diameters
introduces a significant error to the theory of reconstruction as it
changes the geometric measure of probability.
Numerous serious errors have been made in a group of the
reconstruction methods based on the change of the areas of
random sections (methods by Johnson and Saltykov). It is of very
high interest to get the source information on the structure as the
area distribution of the statistical objects because it does not require
assumption concerning the shape of the flat cut grains. However, it
violates the choice of the geometric measure of probability.
Johnson’s method has been introduced into ASTM E112 just for
assessment of parameters of the structure in the flat section, without
a volumetric reconstruction.